Thanks @MichaelHardy, yes there are a couple of tutorial examples in textbooks which cover this. E.g. taking $ \tan^{-1}\left(\frac{ax+1}{a-x}\right)$ to be an antiderivative of $ \frac{1}{1+x^2}$ is interesting, and related. All the examples/discussion I've found so far are from a pure mathematics perspective. I'm interested in applications (including pure mathematics techniques) where different constants are needed.

Thanks @RobertIsrael for suggesting MSE. I'm undertaking an educational research project and whether this technical detail from analysis "matters". It would be helpful to ask in a research forum (i.e. here) whether this opportunity with constants on different connected domains is needed/requires in particular situations/applications. I hope I'm not asking an elementary question in the wrong forum.

Indeed! A point often missed, or quietly passed over in textbooks. My question asks whether this is ever needed or used?

Yes, indeed, and Polya makes a lot of this in his "Induction and Analogy" book.